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Beliebt Rechnen >

integral of 5x^4(x^5+2)^9

Lösung

\int 5 x^{4} (x^{5} + 2)^{9} d x

Lösung

\frac{1}{10} x^{5} (x^{45} + 20 x^{40} + 180 x^{35} + 960 x^{30} + 3360 x^{25} + 8064 x^{20} + 13440 x^{15} + 15360 x^{10} + 11520 x^{5} + 5120) + C

Schritte zur Lösung

\int 5 x^{4} (x^{5} + 2)^{9} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int x^{4} (x^{5} + 2)^{9} d x

Wende U-Substitution an

= 5 \cdot \int \frac{u ^{\frac{1}{4}} ( u ^{\frac{5}{4}} + 2 ) ^{9}}{4} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \frac{1}{4} \cdot \int u^{\frac{1}{4}} (u^{\frac{5}{4}} + 2)^{9} d u

Multipliziere aus

u^{\frac{1}{4}} (u^{\frac{5}{4}} + 2)^{9} : u^{\frac{23}{2}} + 18 u^{\frac{41}{4}} + 144 u^{9} + 672 u^{\frac{31}{4}} + 2016 u^{\frac{13}{2}} + 4032 u^{\frac{21}{4}} + 5376 u^{4} + 4608 u^{\frac{11}{4}} + 2304 u^{\frac{3}{2}} + 512 u^{\frac{1}{4}}

= 5 \cdot \frac{1}{4} \cdot \int u^{\frac{23}{2}} + 18 u^{\frac{41}{4}} + 144 u^{9} + 672 u^{\frac{31}{4}} + 2016 u^{\frac{13}{2}} + 4032 u^{\frac{21}{4}} + 5376 u^{4} + 4608 u^{\frac{11}{4}} + 2304 u^{\frac{3}{2}} + 512 u^{\frac{1}{4}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 \cdot \frac{1}{4} (\int u^{\frac{23}{2}} d u + \int 18 u^{\frac{41}{4}} d u + \int 144 u^{9} d u + \int 672 u^{\frac{31}{4}} d u + \int 2016 u^{\frac{13}{2}} d u + \int 4032 u^{\frac{21}{4}} d u + \int 5376 u^{4} d u + \int 4608 u^{\frac{11}{4}} d u + \int 2304 u^{\frac{3}{2}} d u + \int 512 u^{\frac{1}{4}} d u)

\int u^{\frac{23}{2}} d u = \frac{2}{25} u^{\frac{25}{2}}

\int 18 u^{\frac{41}{4}} d u = \frac{8}{5} u^{\frac{45}{4}}

\int 144 u^{9} d u = \frac{72 u ^{10}}{5}

\int 672 u^{\frac{31}{4}} d u = \frac{384}{5} u^{\frac{35}{4}}

\int 2016 u^{\frac{13}{2}} d u = \frac{1344}{5} u^{\frac{15}{2}}

\int 4032 u^{\frac{21}{4}} d u = \frac{16128}{25} u^{\frac{25}{4}}

\int 5376 u^{4} d u = \frac{5376 u ^{5}}{5}

\int 4608 u^{\frac{11}{4}} d u = \frac{6144}{5} u^{\frac{15}{4}}

\int 2304 u^{\frac{3}{2}} d u = \frac{4608}{5} u^{\frac{5}{2}}

\int 512 u^{\frac{1}{4}} d u = \frac{2048}{5} u^{\frac{5}{4}}

= 5 \cdot \frac{1}{4} (\frac{2}{25} u^{\frac{25}{2}} + \frac{8}{5} u^{\frac{45}{4}} + \frac{72 u ^{10}}{5} + \frac{384}{5} u^{\frac{35}{4}} + \frac{1344}{5} u^{\frac{15}{2}} + \frac{16128}{25} u^{\frac{25}{4}} + \frac{5376 u ^{5}}{5} + \frac{6144}{5} u^{\frac{15}{4}} + \frac{4608}{5} u^{\frac{5}{2}} + \frac{2048}{5} u^{\frac{5}{4}})

Setze in

u = x^{4}

ein

= 5 \cdot \frac{1}{4} (\frac{2}{25} (x^{4})^{\frac{25}{2}} + \frac{8}{5} (x^{4})^{\frac{45}{4}} + \frac{72 ( x ^{4} ) ^{10}}{5} + \frac{384}{5} (x^{4})^{\frac{35}{4}} + \frac{1344}{5} (x^{4})^{\frac{15}{2}} + \frac{16128}{25} (x^{4})^{\frac{25}{4}} + \frac{5376 ( x ^{4} ) ^{5}}{5} + \frac{6144}{5} (x^{4})^{\frac{15}{4}} + \frac{4608}{5} (x^{4})^{\frac{5}{2}} + \frac{2048}{5} (x^{4})^{\frac{5}{4}})

Vereinfache

5 \cdot \frac{1}{4} (\frac{2}{25} (x^{4})^{\frac{25}{2}} + \frac{8}{5} (x^{4})^{\frac{45}{4}} + \frac{72 ( x ^{4} ) ^{10}}{5} + \frac{384}{5} (x^{4})^{\frac{35}{4}} + \frac{1344}{5} (x^{4})^{\frac{15}{2}} + \frac{16128}{25} (x^{4})^{\frac{25}{4}} + \frac{5376 ( x ^{4} ) ^{5}}{5} + \frac{6144}{5} (x^{4})^{\frac{15}{4}} + \frac{4608}{5} (x^{4})^{\frac{5}{2}} + \frac{2048}{5} (x^{4})^{\frac{5}{4}}) : \frac{1}{10} x^{5} (x^{45} + 20 x^{40} + 180 x^{35} + 960 x^{30} + 3360 x^{25} + 8064 x^{20} + 13440 x^{15} + 15360 x^{10} + 11520 x^{5} + 5120)

= \frac{1}{10} x^{5} (x^{45} + 20 x^{40} + 180 x^{35} + 960 x^{30} + 3360 x^{25} + 8064 x^{20} + 13440 x^{15} + 15360 x^{10} + 11520 x^{5} + 5120)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{10} x^{5} (x^{45} + 20 x^{40} + 180 x^{35} + 960 x^{30} + 3360 x^{25} + 8064 x^{20} + 13440 x^{15} + 15360 x^{10} + 11520 x^{5} + 5120) + C

Graph

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